A string is 1 metre & formed into a rectangle with 1 pair of opp. sides each x cm.Find value of x
which'll maximise the area enclosed by the string.
Solution
If two opposite sides are x cm each then the other opposite side would be;
[y = (1-2x)/2]
If the area of the rectangle is A;
[A = x*y]
[A = x((1-2x)/2)]
[A = (1/2)(x-2x^2)]
For area A to be a maximum or minimum [(dA)/dx = 0]
If it is a maximum then [(d^2A)/dx<0] at the point where [(dA)/dx = 0.]
[(dA)/dx ]
[= (1/2)(1-2x)]
[(dA)/dx = 0 ]
[(1/2)(1-2x) = 0]
[ x = 1/2]
[(d^2A)/dx = (1/2)(-2) = -1]
Always [(d^2A)/dx<0]
Therefore at x = 0.5cm we have the maximum area.
When x = 0.5 cm the rectangle will become a square.